Distributionally robust stochastic variational inequalities

Mathematical Programming - We propose a formulation of the distributionally robust variational inequality (DRVI) to deal with uncertainties of distributions of the involved random variables in...     

Distributionally robust stochastic shortest path problem

This paper considers a stochastic version of the shortest path problem, the Distributionally Robust Stochastic Shortest Path Problem(DRSSPP) on directed graphs. In this model, the arc costs are deterministic, while each arc has a random delay. The mean vector and the second-moment matrix of the uncertain data are assumed known, but the exact information of the distribution is unknown. A penalty occurs when the given delay constraint is not satisfied. The objective is to minimize the sum of the path cost and the expected path delay penalty. As it is NP-hard, we approximate the DRSSPP with a semidefinite programming (SDP for short) problem, which is solvable in polynomial time and provides tight lower bounds.     

Distributionally robust stochastic programs with side information based on trimmings

Mathematical Programming - We consider stochastic programs conditional on some covariate information, where the only knowledge of the possible relationship between the uncertain parameters and the...     

Concerning a stochastic dynamical system

We study the discrete-time dynamical system $$X_{n + 1} = 2\sigma \cos (2\pi \theta _n )g(X_n ), n \in \mathbb{Z},$$ Whereθ _n is an ergodic stationary process whose univariate distribution is uniform on the interval [0, 1], the functiong(x) is odd, bounded, increasing, and continous, and ? is the ring of integers. It is proved that under certain conditions there exists a unique stationary process that is a solution of the above equation and this process has a continous purely singular spectrum.     

Sensitivity analysis and parameter identification for a nonlinear time-delay system in microbial fed-batch process

Developing suitable dynamic models of bioprocess is a difficult issue in bioscience. In this paper, considering the microbial metabolism mechanism, i.e., the production of new biomass is delayed by the amount of time it takes to metabolize the nutrients, in glycerol bioconversion to 1,3-propanediol, we propose a nonlinear time-delay system to formulate the fed-batch fermentation process. Some important properties are also discussed. Then, in view of the effect of time-delay and the high number of kinetic parameters in the system, the parametric sensitivity analysis is used to determine the key parameters. Finally, a parameter identification model is presented and a global optimization method is developed to seek the optimal key parameters. Numerical results show that the nonlinear time-delay system can describe the fed-batch fermentation process reasonably.     

Rapid parameter identification of linear time-delay system from noisy frequency domain data

The technique to identify the system parameters thereof has attracted extensive research interest, since knowing the parameters would enable effective system control strategy and accurate response prediction. In this paper, a novel approach is developed to identify the parameters of the linear time-delay differential system by analyzing the complex system response in the frequency domain. Firstly, the complex frequency response of the time-delay system is expressed as a function of physical parameters and time-delay parameters, forming a typical optimization problem. Subsequently, the sensitivities with respect to the unknown parameters are derived. A novel sensitivity-based algorithm is adopted in the identification procedure. Trust-region constraint is implemented and hence tackled by Tikhonov regularization, which effectively enhances the efficiency of the algorithm. The feasibility and robustness of the identification procedure are evaluated by identifying the parameters of two numerical time-delay systems and an experimental case.     

Distributionally robust discrete optimization with Entropic Value-at-Risk

We study the discrete optimization problem under the distributionally robust framework. We optimize the Entropic Value-at-Risk, which is a coherent risk measure and is also known as Bernstein approximation for the chance constraint. We propose an efficient approximation algorithm to resolve the problem via solving a sequence of nominal problems. The computational results show that the number of nominal problems required to be solved is small under various distributional information sets.     

Fast-slow stochastic dynamical system with singular coefficients

This paper aims to study the asymptotic behavior of a fast-slow stochastic dynamical system with singular coefficients, where the fast motion is given by a continuous diffusion process while the slow component is driven by an α-stable noise with α ∈ [1, 2). Using Zvonkin’s transformation and the technique of the Poisson equation, we have that both the strong and weak convergences in the averaging principle are established, which can be viewed as a functional law of large numbers. Then we study t...     

Distributionally robust modeling of optimal control

The aim of this paper is to formulate several questions related to distributionally robust Stochastic Optimal Control modeling. As an example the distributionally robust counterpart of the classical inventory model is discussed in details. Finite and infinite horizon stationary settings are considered.     

A distributionally robust optimization model for batch nonlinear switched time-delay system considering uncertain output measurements

In this paper, we consider a nonlinear switched time-delay (NSTD) system with unknown switching times and unknown system parameters, where the output measurement is uncertain. This system is the underling dynamical system for the batch process of glycerol bioconversion to 1,3-propanediol induced by Klebsiella pneumoniae. The uncertain output measurement is regarded as a stochastic vector (whose components are stochastic variables) and the only information about its distribution is the first-order moment. The objective of this paper is to identify the unknown quantities of the NSTD system. For this, a distributionally robust optimization problem (a bi-level optimization problem) governed by the NSTD system is proposed, where the relative error under the environment of uncertain output measurements is involved in the cost functional. The bi-level optimization problem is transformed into a single-level optimization problem with non-smooth term through the application of duality theory in probability space. By applying the smoothing technique, the non-smooth term is approximated by a smooth term and the convergence of the approximation is established. Then, the gradients of the cost functional with respect to switching times and system parameters are derived. A hybrid optimization algorithm is developed to solve the transformed problem. Finally, we verify the obtained switching times and system parameters, as well as the effectiveness of the proposed algorithm, by solving this distributionally robust optimization problem.     

Non-fragile robust stabilization and control for uncertain stochastic nonlinear time-delay systems

This paper deals with the problem of non-fragile robust stabilization and H_∞ control for a class of uncertain stochastic nonlinear time-delay systems. The parametric uncertainties are real time-varying as well as norm bounded. The time-delay factors are unknown and time-varying with known bounds. The aim is to design a memoryless non-fragile state feedback control law such that the closed-loop system is stochastically asymptotically stable in the mean square and the effect of the disturbance input on the controlled output is less than a prescribed level for all admissible parameter uncertainties. New sufficient conditions for the existence of such controllers are presented based on the linear matrix inequalities (LMIs) approach. Numerical example is given to illustrate the effectiveness of the developed techniques.     

Vibration suppression of a dynamical system to multi-parametric excitations via time-delay absorber

The aim of this work is to control the dynamic system behavior represented by a beam at simultaneous primary and sub-harmonic resonance condition, where the system damage is probable. Control is conducted via time delay absorber to suppress chaotic vibrations. A comprehensive investigation of the effect of the time delay on the control of a beam when subjected to multi- parametric excitation forces is presented. Multiple scale perturbation method is applied to obtain the solution up to the second order approximation. Different resonance cases are reported and studied numerically. Stability of the steady state solution for the selected resonance case is investigated applying Rung-Kutta fourth order method and frequency response equations via Matlab 7.0 and Maple11. Time delay absorber is effective like ordinary one within a specified range of time delay. The delay time is an important factor in selecting the absorber. The effects of the different parameters of the absorber on the system behavior are studied numerically. The reported results are compared with the available published work.     

Synchronization for a class of complex dynamical networks with time-delay

In this study, the synchronization problem is addressed for a class of complex dynamical networks in which every identical node is a time-delayed Lur’e system. Delay-dependent and delay-independent synchronization criteria are established through a decoupling technique, which reduces a group of high-dimensional linear matrix inequalities (LMIs) to the test of two groups of lower-dimensional LMIs. An extension to the synchronization of discrete-time Lur’e networks with time delay is also studied. The efficiency and applicability of the proposed methodology is demonstrated by a numerical example through simulation.     

Distributionally robust chance constraint with unimodality-skewness information and conic reformulation

Moment-based ambiguity sets are mostly used in distributionally robust chance constraints (DRCCs). Their conservatism can be reduced by imposing unimodality, but the known reformulations do not scale well. We propose a new ambiguity set tailored to unimodal and seemingly symmetric distributions by encoding unimodality-skewness information, which leads to conic reformulations of DRCCs that are more tractable than known ones based on semi-definite programs. Besides, the conic reformulation yields a closed-form expression of the inverse of unimodal Cantelli's bound.     

Nonnegative compartment dynamical system modelling with stochastic differential equations

Compartment models are widely used in various physical sciences and adequately describe many biological phenomena. Elements such as blood, gut, liver and lean tissue are characterized as homogeneous compartments, within which the drug resides for a time, later to transit to another compartment, perhaps recycling or eventually vanishing. We address the issue of compartment dynamical system modelling using multidimensional stochastic differential equations, rather than the classical approach based on the continuous-time Markov chain. Pure-jump processes are employed as perturbation to the deterministic compartmental dynamical system. Unlike with the Brownian motion, noise can be incorporated into both outflows and inter-compartmental flows without violating nonnegativity of the compartmental system, under mild technical conditions. The proposed formulation is easy to simulate, shares various essential properties with the corresponding deterministic ordinary differential equation, such as asymptotic behaviors in mean, steady states and average residence times, and can be made as close to the corresponding diffusion approximation as desired. Asymptotic mean-square stability of the steady state is proved to hold under some assumptions on the model structure. Numerical results are provided to illustrate the effectiveness of our formulation.     

Distributionally robust optimization with correlated data from vector autoregressive processes

We present a distributionally robust formulation of a stochastic optimization problem for non-i.i.d vector autoregressive data. We use the Wasserstein distance to define robustness in the space of distributions and we show, using duality theory, that the problem is equivalent to a finite convex–concave saddle point problem. The performance of the method is demonstrated on both synthetic and real data.     

Distributionally robust mixed integer linear programs: Persistency models with applications

In this paper, we review recent advances in the distributional analysis of mixed integer linear programs with random objective coefficients. Suppose that the probability distribution of the objective coefficients is incompletely specified and characterized through partial moment information. Conic programming methods have been recently used to find distributionally robust bounds for the expected optimal value of mixed integer linear programs over the set of all distributions with the given moment information. These methods also provide additional information on the probability that a binary variable attains a value of 1 in the optimal solution for 0–1 integer linear programs. This probability is defined as the persistency of a binary variable. In this paper, we provide an overview of the complexity results for these models, conic programming formulations that are readily implementable with standard solvers and important applications of persistency models. The main message that we hope to convey through this review is that tools of conic programming provide important insights in the probabilistic analysis of discrete optimization problems. These tools lead to distributionally robust bounds with applications in activity networks, vertex packing, discrete choice models, random walks and sequencing problems, and newsvendor problems.     

Graph problems arising from parameter identification of discrete dynamical systems

This paper focuses on combinatorial feasibility and optimization problems that arise in the context of parameter identification of discrete dynamical systems. Given a candidate parametric model for a physical system and a set of experimental observations, the objective of parameter identification is to provide estimates of the parameter values for which the model can reproduce the experiments. To this end, we define a finite graph corresponding to the model, to each arc of which a set of parameters is associated. Paths in this graph are regarded as feasible only if the sets of parameters corresponding to the arcs of the path have nonempty intersection. We study feasibility and optimization problems on such feasible paths, focusing on computational complexity. We show that, under certain restrictions on the sets of parameters, some of the problems become tractable, whereas others are NP-hard. In a similar vein, we define and study some graph problems for experimental design, whose goal is to support the scientist in optimally designing new experiments.